Review: Potential stumbling blocks…

Whether the negation sign is on the inside or the outside of

a quantified statement makes a big difference!

Example: Let T(x) ≡ “x is tall”. Consider the following:

� ¬ ∀ x T(x)

� “It is not the case that all people are tall.”

� ∀ x ¬T(x)

� “For all people x, it is not the case that x is tall.”

Note: ¬ ∀ x T(x) = ∃ x ¬T(x) ≠ ∀ x ¬T(x)

Recall: When we push negation into a quantifier, DeMorgan’s law says that we need to switch the quantifier!

Review: Potential stumbling blocks…

Let: C(x) ≡ “x is enrolled in CS441”

S(x) ≡ “x is smart.”

Question: The following two statements look the same, what’s the difference?

� ∃ x [C(x) ∧∧∧∧ S(x)]

� ∃ x [C(x) → S(x)]

Subtle note: The second statement is true if there exists even one smart person on Earth, because F→T.

There exists a student x such that if x is in CS441, then x is

smart.

There is a smart student in CS441.

Review: Translation

� Suppose:

� Variables x,y denote people

� L(x,y) denotes "x loves y"

� Translate:

� Everybody loves Raymond

� Everybody loves somebody.

� There is somebody whom everybody loves.

� There is somebody who Raymond doesn't love.

� There is somebody whom no one loves.

� Everybody loves himself.

Review: Evaluate

� Domain of discourse = positive integers.

� Let Q(x,y) denote x*x = 2*y

� Let T(x,y) denote x^2 = x*y

� Q(4,8)

� ∃x Q(x,50)

� ∀x Q(x,x)

� ∃x ∃y Q(x,y)

� ∃x ∀y Q(x,y)

� ∀x ∃y Q(x,y)

� ∀x ∀y Q(x,y)

Today’s topic

� Rules of inference

What have we learned? Where are we going?

Propositional logic (representation)

Predicate logic (refined

representation) Quantifiers (generalization)

Inference and proof (deriving new knowledge!)

Writing valid proofs is a subtle art

Step 1: Discover and

formalize the property

that you wish to prove Step 2: Formalize the ground truths

(axioms) that you will use to prove

this property

Step 3: Show that the property in

question follows from the truth of

your axioms

This is called

“research”

Subtle, but not terribly difficult

Oh, the errors you will make ☺☺☺☺

What is science without jargon?

A conjecture is a statement that is thought to be true.

A proof is a valid argument that establishes the truth

of a given statement (i.e., a conjecture)

After a proof has been found for a given conjecture, it

becomes a theorem

A sequence of statements ending with a conclusion

The truth of the conclusion

follows from the truth of the preceding statements

A tale of two proof techniques

In a formal proof, each step of

the proof clearly follows from

the postulates and axioms

assumed in the conjecture.

In an informal proof, one step in

the proof may consist of

multiple derivations, portions of

the proof may be skipped or

assumed correct, and axioms

may not be explicitly stated.

Statements that are assumed to be true

Consider the following argument:

“If you have an account, you can access the network”

“You have an account”

Therefore,

“You can access the network”Premises

Conclusion

This argument seems valid, but how can we

demonstrate this formally??

How can we formalize an argument?

Let’s analyze the form of our argument

“If you have an account, then you can access the network”

“You have an account”

Therefore,

“You can access the network”

p → q

p

∴ q

This is called a

“rule of inference”

p q

Rules of inference allow us to make valid arguments

� Many times, we can determine whether an

argument is valid by using a truth table, but this is

often a cumbersome approach

� Instead, we can apply a sequence of rules of

inference to draw valid conclusions from a set of

premises

p → q

p

∴ q

Let’s analyze the form of our argument

“If you have an account, you can access the network”

“You have an account”

Therefore,

“You can access the network”p → q

p

∴ qThis form is equivalent to the statement

((p → q) ∧∧∧∧ p) → q

Since ((p → q) ∧∧∧∧ p) → q is a tautology, we

know that our argument is valid!

Rules of inference are logically valid ways to draw

conclusions when constructing a formal proof

The previous rule is called modus ponens

� Rule of inference:

� Informally: Given an implication p → q, if we know that p

is true, then q is also true

But why can we trust modus ponens?

� Tautology: ((p → q) ∧∧∧∧ p) → q

� Truth table:

p → q

p

∴ q

p q p→q

T T T

T F F

F T T

F F T

Any time that p→q

and p are both true, q is also true!

There are lots of other rules of inference that

we can use!

Addition

� Tautology: p → (p V q)

� Rule of inference:

� Example: “It is raining now, therefore it is raining now or

it is snowing now.”

Simplification

� Tautology: p ∧∧∧∧ q → p

� Rule of inference:

� Example: “It is cold outside and it is snowing. Therefore, it

is cold outside.”

p

∴ p V q

p ∧ q

∧ p

There are lots of other rules of inference that

we can use!Modus tollens

� Tautology: [¬q ∧∧∧∧ (p → q)] → ¬p

� Rule of inference:

� Example: “If I am hungry, then I will eat. I am not eating.

Therefore, I am not hungry.”

Hypothetical syllogism

� Tautology: [(p → q) ∧∧∧∧ (q → r)] → (p → r)

� Rule of inference:

� Example: “If I eat a big meal, then I feel full. If I feel full,

then I am happy. Therefore, if I eat a big meal, then I am

happy.”

p → q

¬q

∧ ¬p

(p → q)

(q → r)

∧ (p → r)

There are lots of other rules of inference that

we can use!Disjunctive syllogism

� Tautology: [¬p ∧∧∧∧ (p V q)] → q

� Rule of inference:

� Example: “Either the heat is broken, or I have a fever.

The heat is not broken, therefore I have a fever.”

Conjunction

� Tautology: [(p) ∧∧∧∧ (q)] → (p ∧∧∧∧ q)

� Rule of inference:

� Example: “Jack is tall. Jack is skinny. Therefore, Jack is

tall and skinny.”

p ∧ q

¬p

∧ q

p

q

\ (p ∧ q)

There are lots of other rules of inference that

we can use!Resolution

� Tautology: [(p V q) ∧∧∧∧ (¬p V r)] → (q V r)

� Rule of inference:

� Example: “If it is not raining, I will ride my bike. If it is

raining, I will lift weights. Therefore, I will either ride my

bike or lift weights”

Special cases:

1. If r = q, we get

2. If r = F, we get

p ∧ q

¬p ∧ r

∧ q ∧ r

p ∧ q

¬p ∧ q

∧ q

p ∧ q

¬p

∧ q

We can use rules of inference to build valid arguments

If it is raining, I will stay inside. If am inside,

Stephanie will come over. If Stephanie comes over

and it is a Saturday, then we will play Scrabble. Today

is Saturday. It is raining.

Let:

� r ≡ It is raining

� i ≡ I am inside

� s ≡ Stephanie will come over

� c ≡ we will play Scrabble

� a ≡ it is Saturday

Hypotheses:

� r → i

� i → s

� s ∧ a → c

� a

� r

We can use rules of inference to build valid arguments

Let:

� r ≡ It is raining

� i ≡ I am inside

� s ≡ Stephanie will come over

� c ≡ we will play Scrabble

� a ≡ it is Saturday

Step:

1. r → i hypothesis

2. i → s hypothesis

3. r → s hypothetical syllogism with 1 and 2

4. r hypothesis

5. s modus ponens with 3 and 4

6. a hypothesis

7. s ∧∧∧∧ a conjunction of 5 and 6

8. s ∧∧∧∧ a → c hypothesis

9. c modus ponens with 7 and 8

Hypotheses:

� r → i

� i → s

� s ∧∧∧∧ a → c

� a

� r

I will playScrabble!

We also have rules of inference for statements

with quantifiers

Universal Instantiation

� Intuition: If we know that P(x) is true for all x, then P(c) is

true for a particular c

� Rule of inference:

Universal Generalization

� Intuition: If we can show that P(c) is true for an arbitrary c,

then we can conclude that P(x) is true for any x

� Rule of inference:

∧x P(x)

∧ P(c)

P(c)

∧ ∧xP(x)

Note that “arbitrary” does not mean “randomly chosen.” It means that we cannot make any

assumptions about c other than the fact that it comes

from the appropriate domain.

We also have rules of inference for statements

with quantifiers

Existential Instantiation

� Intuition: If we know that ∃ P(x) is true, then we know that

P(c) is true for some c

� Rule of inference:

Existential Generalization

� Intuition: If we can show that P(c) is true for a particular c,

then we can conclude that ∃ P(x) is true

� Rule of inference:

∧x P(x)

∧ P(c)

P(c)

∧ ∧xP(x)

Again, we cannot make assumptions about c other

than the fact that it exists and is from the appropriate domain.

Hungry dogs redux

Given: All of my dogs like peanut butter

Given: Kody is one

of my dogs

M(x) P(x)

1. ∀ x [M(x) → P(x)] hypothesis

2. M(Kody) hypothesis

3. M(Kody) → P(Kody) universial instantiation from 1

4. P(Kody) modus ponens from 2 and 3

M(Kody)

Reasoning about our class

Show that the premises “A student in this class has not

read the book” and “everyone in this class turned in

HW1” imply the conclusion “Someone who turned in

HW1 has not read the book.”

Let:

� C(x) ≡ x is in this class

� B(x) ≡ x has read the book

� T(x) ≡ x turned in HW1

Premises:

� ∃ x [C(x) ∧∧∧∧ ¬B(x)]

� ∀ x [C(x) → T(x)]

Reasoning about our class

Let:

� C(x) ≡ x is in this class

� B(x) ≡ x has read the book

� T(x) ≡ x turned in HW1

Steps:

1. ∃ x [C(x) ∧∧∧∧ ¬B(x)]hypothesis

2. C(a) ∧∧∧∧ ¬B(a) existential instantiation from 1

3. C(a) simplification from 2

4. ∀ x [C(x) → T(x)]hypothesis

5. C(a) → T(a) universal instantiation from 4

6. T(a) modus ponens from 5 and 3

7. ¬B(a) simplification from 2

8. T(a) ∧∧∧∧ ¬B(a) conjunction of 6 and 7

9. ∃ x [T(x) ∧∧∧∧ ¬B(x)]existential generalization from 8

Premises:

� ∃ x [C(x) ∧∧∧∧ ¬B(x)]

� ∀ x [C(x) → T(x)]

Group work!

Problem 1: Which rules of inference were used to

make the following arguments?

� Kangaroos live in Australia and are marsupials. Therefore,

kangaroos are marsupials.

� Linda is an excellent swimmer. If Linda is an excellent

swimmer, then she can work as a lifeguard. Therefore,

Linda can work as a lifeguard.

Problem 2: Show that the premises “Everyone in this

discrete math class has taken a course in computer

science” and “Melissa is a student in this discrete

math class” lead to the conclusion “Melissa has taken

a course in computer science.”

We can’t always use formal proof techniques

Result: Most mathematical proofs are actually constructed using informal proof techniques!

Formal proofs are precise

and “easy” for machines

to construct…

… but are often tedious for

humans to construct,

interpret, or verify.

Another Example

� 1. It is not sunny this afternoon and it is colder than

yesterday.

� 2. We will go swimming only if it is sunny.

� 3. If we do not go swimming then we will take a

canoe trip.

� 4. If we take a canoe trip, then we will be home by

sunset.

� Prove: We will be home by sunset.

What are the characteristics of an informal proof?

In an informal proof

� The statements making up the proof are typically not

written in any formal language (e.g., propositional logic)

� Steps of the proof and derivations are often argued using

English or mathematical formulas

� Multiple derivations may occur in a single step

� Axioms are often not all stated up front

As a result, it is sometimes easy to make mistakes

writing informal proofs.

Final Thoughts

� Until today, we had look at representing different

types of logical statements

� Rules of inference allow us to derive new results by

reasoning about known truths

� Next lecture: � Proof techniques